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A minimum residual method (the method of key shifts) is proposed for the structure determination of superstructures. The method is based on the fact that there are in most cases only a few important changes (key shifts) of atomic parameters in the extension from a substructure to the corresponding superstructure. The method starts from the substructure. We change each atomic parameter of each atom by particular amounts in the plus and minus directions, and calculate the corresponding  R  values. The key shifts can then be detected from the steepest descents of the  R  values. Once an approximate superstructure has been found out by this method, i.e. approximately correct phases have been assigned to the reflections, the structure can be refined by ordinary Fourier and least-squares methods. The method has successfully been applied to the structure determination of freieslebenite, PbAgSbS 3  and jordanite, Pb 28− x  As 12 S 46− x  .

On the application of a minimum residual method to the structure determination of superstructures*